How to create a plane about such dimensions as the length of other edge?
There are several ways to model with precision, and the first step in this kind of modeling will be to turn on Edge Length and / or Face Area options in 3D View > T panel > Mesh Display rollout. 



After duplicating the original edge there are several options to model square plane then.

Extrude with snapping


Turn on snapping with Shift+Tab and set it to Increment. Alternatively, without turning it on hold Ctrl while extruding.
Extrude edge and see the values of the edges.




Duplicate and rotate original edge


Set the Pivot point to Active Element, duplicate original edge, rotate it by 90 degrees around appropriate axis (in this case Z).
Select all and press W > Remove doubles (because one appeared when you rotated the edge).
Select 2 vertices on the edges' ends and connect them with F. The triangle was created.
Subdivide the diagonal edge. Select the vertex which apeared after subdivisioun, turn on snapping, set it to Vertex and grab selected vertex to align it by X and Y axes with already existing vertices.




Read:


Showing measures in the viewport
How to model effectively using exact measurements - related question.
Mesh Display panel - wiki